A circuit takes a current of `3 A` at a power factor of 0.6 lagging when connected. to a `115 V- 50 Hz` supply. Another circuit takes a current of `5 A` at a power factor of `0.707` leading when connected to the same supply. If the two circuits are connected in series across a `230 V ,50 Hz` supply, then calculate
(a) the current
(b) the power consumed and (c) the power factor

Explanation

To solve the problem step by step, we will analyze each circuit separately, calculate their impedances, resistances, and then combine them when they are connected in series.

### Step 1: Analyze Circuit 1
**Given:**
- Current \( I_1 = 3 \, A \)
- Power Factor \( \text{pf}_1 = 0.6 \) (lagging)
- Voltage \( V_1 = 115 \, V \)

**Calculate Impedance \( Z_1 \):**
\[ Z_1 = \frac{V_1}{I_1} = \frac{115}{3} = 38.33 \, \Omega \]

**Calculate Resistance \( R_1 \):**
\[ R_1 = Z_1 \cdot \text{pf}_1 = 38.33 \cdot 0.6 = 23 \, \Omega \]

**Calculate Inductive Reactance \( X_L \):**
\[ X_L = \sqrt{Z_1^2 - R_1^2} = \sqrt{(38.33)^2 - (23)^2} = 30.67 \, \Omega \]

### Step 2: Analyze Circuit 2
**Given:**
- Current \( I_2 = 5 \, A \)
- Power Factor \( \text{pf}_2 = 0.707 \) (leading)
- Voltage \( V_2 = 115 \, V \)

**Calculate Impedance \( Z_2 \):**
\[ Z_2 = \frac{V_2}{I_2} = \frac{115}{5} = 23 \, \Omega \]

**Calculate Resistance \( R_2 \):**
\[ R_2 = Z_2 \cdot \text{pf}_2 = 23 \cdot 0.707 = 16.26 \, \Omega \]

**Calculate Capacitive Reactance \( X_C \):**
\[ X_C = \sqrt{Z_2^2 - R_2^2} = \sqrt{(23)^2 - (16.26)^2} = 16.26 \, \Omega \]

### Step 3: Combine Circuits in Series
**Total Resistance \( R \):**
\[ R = R_1 + R_2 = 23 + 16.26 = 39.26 \, \Omega \]

**Net Reactance \( X \):**
\[ X = X_L - X_C = 30.67 - 16.26 = 14.41 \, \Omega \]

**Calculate Total Impedance \( Z \):**
\[ Z = \sqrt{R^2 + X^2} = \sqrt{(39.26)^2 + (14.41)^2} = 41.82 \, \Omega \]

### Step 4: Calculate Current in Series Circuit
**Given Supply Voltage \( V = 230 \, V \):**
\[ I = \frac{V}{Z} = \frac{230}{41.82} = 5.5 \, A \]

### Step 5: Calculate Power Consumed
**Power Consumed \( P \):**
\[ P = I^2 \cdot R = (5.5)^2 \cdot 39.26 = 1187.6 \, W \]

### Step 6: Calculate Power Factor
**Power Factor \( \text{pf} \):**
\[ \text{pf} = \frac{R}{Z} = \frac{39.26}{41.82} = 0.939 \]

### Summary of Results
(a) Current \( I = 5.5 \, A \)
(b) Power Consumed \( P = 1187.6 \, W \)
(c) Power Factor \( \text{pf} = 0.939 \) (lagging)